{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,19]],"date-time":"2026-02-19T08:07:11Z","timestamp":1771488431582,"version":"3.50.1"},"reference-count":31,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2021,3,24]],"date-time":"2021-03-24T00:00:00Z","timestamp":1616544000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100004239","name":"Politechnika Pozna\u0144ska","doi-asserted-by":"publisher","award":["0211\/SBAD\/0120"],"award-info":[{"award-number":["0211\/SBAD\/0120"]}],"id":[{"id":"10.13039\/501100004239","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>We consider series systems built of components which have independent identically distributed (iid) lifetimes with an increasing failure rate (IFR). We determine sharp upper bounds for the expectations of the system lifetimes expressed in terms of the mean, and various scale units based on absolute central moments of component lifetimes. We further establish analogous bounds under a more stringent assumption that the component lifetimes have an increasing density (ID) function. We also indicate the relationship between the IFR property of the components and the generalized cumulative residual entropy of the series system lifetime.<\/jats:p>","DOI":"10.3390\/e23040385","type":"journal-article","created":{"date-parts":[[2021,3,24]],"date-time":"2021-03-24T15:42:19Z","timestamp":1616600539000},"page":"385","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Bounds on the Lifetime Expectations of Series Systems with IFR Component Lifetimes"],"prefix":"10.3390","volume":"23","author":[{"given":"Tomasz","family":"Rychlik","sequence":"first","affiliation":[{"name":"Institute of Mathematics, Polish Academy of Sciences, \u015aniadeckich 8, 00656 Warsaw, Poland"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Magdalena","family":"Szymkowiak","sequence":"additional","affiliation":[{"name":"Institute of Automation and Robotics, Poznan University of Technology, Plac Marii Sk\u0142odowskiej-Curie 5, 60965 Poznan, Poland"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,3,24]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"163","DOI":"10.1016\/j.ijpe.2017.10.022","article-title":"Reliability optimization for series systems under uncertain component failure rates in the design phase","volume":"196","author":"Ge","year":"2018","journal-title":"Int. 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